Chapter 28: Mathematical Applications 
TABLE 28-5. 
Calculation of expected returns. 
Price of XYZ in 6 Months 
Below 30 
31 
32 
33 
34 
Above 35 
467 
Chance of XYZ Being at That Price · 
20% 
10% 
10% 
10% 
10% 
40% 
100% 
Since the percentages total 100%, all the outcomes have theoretically been 
allowed for. Now suppose a February 30 call is trading at 4 and a February 35 call is 
trading at 2 points. A bull spread could be established by buying the February 30 and 
selling the February 35. This position would cost 2 points - that is, it is a 2-point 
debit. The spreader could make 3 points if XYZ were above 35 at expiration for a 
return of 150%, or he could lose 100% if XYZ were below 30 at expiration. The 
expected return for this spread can be computed by multiplying the outcome at expi­
ration for each price by the probability of being at that price, and then summing the 
results. For example, if XYZ is below 30 at expiration, the spreader loses $200. It was 
assumed that there is a 20% chance of XYZ being below 30 at expiration, so the 
expected loss is 20% times $200, or $40. Table 28-6 shows the computation of the 
expected results at all the prices. The total expected profit is $100. This means that 
the expected return (profit divided by investment) is 50% ($100/$200). This appears 
to be an attractive spread, because the spreader could "expect" to make 50% of his 
money, less commissions. 
What has really been calculated in this example is merely the return that one 
would expect to make in the long run if he invested in the same position many times 
throughout history. Saying that a particular position has an expected return of 8 or 
9% is no different from saying that common stocks return 8 or 9% in the long run. 
Of course, in bull markets stock would do much better, and in bear markets much 
worse. In a similar manner, this example bull spread with an expected return of 50% 
may do as well as the maximum profit or as poorly as losing 100% in any one case. It 
is the total return on many cases that has the expected return of 50%. Mathematical 
theory holds that, if one constantly invests in positions with positive expected returns, 
he should have a better chance of making rrwney.